In nine flights between 1993 and 2002, the Balloon Borne Experiment with a Superconducting Spectrometer (BESS) has measured the energy spectrum of cosmic-ray antiprotons between 0.18 and 4.20 GeV, and the spectra of protons and helium to several hundred GeV. BESS has also placed stringent upper limits on the existence of antihelium and antiduterons. Above about 1 GeV, models in which antiprotons are secondary products of the interactions of primary cosmic rays with the ISM agree with the BESS spectrum. Below 1 GeV, BESS data suggest the presence of an additional source of antiprotons. The antiproton/proton ratios measured between 1993 and 1999, during the Sun's positive-polarity phase, are consistent with simple models of solar modulation. Results from the 2000 flight, following the solar magnetic field reversal, show a sudden increase in the antiproton/proton ratio and tend to favor a charge-sign-dependent drift model. To extend BESS measurements to lower energies, a new instrument, BESS-Polar, is under construction for a flight from Antarctica in 2004.